Finnish pre-service teachers' and upper secondarystudents' understanding of division and reasoningstrategies used

摘要

In this paper, we focus on Finnish pre-service elementary teachers' (N=269) andupper secondary students' (N=1,434) understanding of division. In the questionnaire, weused the following non-standard division problem: "We know that 498:6=83. How couldyou conclude from this relationship (without using long-division algorithm) what 491:6=?is?" This problem especially measures conceptual understanding, adaptive reasoning, andprocedural fluency. Based on the results, we can conclude that division seems not to befully understood: 45% of the pre-service teachers and 37% of upper secondary studentswere able to produce complete or mainly correct solutions. The reasoning strategies used bythese two groups did not differ very much. We identified four main reasons for problems inunderstanding this task: (1) staying on the integer level, (2) an inability to handle theremainder, (3) difficulties in understanding the relationships between different operations,and (4) insufficient reasoning strategies. It seems that learners' reasoning strategies inparticular play a central role when teachers try to improve learners' proficiency.